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On the spectrum of the multiplicative Hilbert matrix

Perfekt, K.-M. and Pushnitski, A. (2018) On the spectrum of the multiplicative Hilbert matrix. Arkiv för Matematik, 56 (1). pp. 163-183. ISSN 1871-2487

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To link to this item DOI: 10.4310/ARKIV.2018.v56.n1.a10

Abstract/Summary

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries (mn−−−√log(mn))−1 for m,n≥2. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with [0,π]. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:72235
Publisher:Royal Swedish Academy of Sciences

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